http://www.math.chalmers.se/Math/Grundutb/CTH/mallar/Style/mven.png

MMA120, Functional Analysis, Autumn 18

  Latest news

 

   Welcome to the course! The schedule for the course can be found in TimeEdit.

 

             

Teachers

   Examiner and lecturer: Genkai Zhang, room MVH 5023, tel: 7725385,

     e-mail: genkai@chalmers.se               

 

 Course literature

        Manfred Einsiedler and Thomas Ward,       Functional Analysis, Spectral Theory and Applications, Springer, 2017.

 

    Chapt. 2, 2.1—2.2. 2.4. (Section 2.2 on space of continuous functions and Stone - Weierstrass theorem is covered  by the course Real Analysis, MMA120, by Ulla Dinger)

 

    Chapt. 3,  3.1-3.2.

    Chapt.  4, 4.1-4.2.  Chapt. 8. 4.1 (Banach-Alaoglu theorem), 4.2  (Applications of B-A theorem, part of)

     Chapt. 5, 5,1.

     Chapt. 7, 7.1-7.4.

    

  Program

      Preliminary Planning:

     Weeks (W) 45-60, Week 1, totally  7 weeks and 2x3 hours per week. Every second Friday morning will be partly exercise class.

      W-I , Chapt. 2. Norms and Banach spaces. (Skip 2.4.1, 2.4.2).

       II, Chapt. 2,  3, 5, Hilbert spaces, Riesz Theorem. Sobolev spaces, Sobolev Embeddings

       III, Chapt. 5, 4, Uniform Boundedness and Open Mappings

        IV, Chapt. 4. 7, Hahn-Banach Thm, L^p and their duals.

              Chapt. 8.1 Banach-Alaoglu theorem

        V, Chapt. 7,  Riesz representations for dual of C(X)

        VI.  Chapt. 8.2 Applications of B-A theorem on the dual of C(X) (as  space of  probability measures.) on equi-distributions: Compactness, convexity, invariant measures/ergodicity.

       

        VII. Repetition.

EXERCISES

      (There are so called Essential Exercises in the book, please work out them. Exercises with * might be a bit difficult)

      2.3a), 2.7, 2.9, 2.16, 2.18, 2.25, 2.26, 2.36*, 2.55, 2.56, 2.58

      3.4, 3.5, 3.9, 3.10, 3.14 (1)(2), 3. 15 (Do this for R^n with l^p-norm first). 3.20, 3.27, 3.28*, 3.37. 3.40

      4.4, 4.5, 4.16, 4.18, 4.20, 4.23, 4.29.   7.2, 7.7, 7.11, 7.33 (a)-(d),   

      7.56, 7.58.           8.6, 8.8, 8.9, 8.16, 8.17, 8.18.

 

 

 

     Assignments

       Homeworks to be handed in. : Exercise and Assignment 1

                                                         Ex. & Assignment 2

                                                          Ex. & Assignment 3

                                                           Ex. & Assignment 4

                                                            Ex. & Assignment 5

 

 

      Examination.

        Hand-in exercises and  oral exam.   Hand-in exercises and  oral exam. There will be 5 assignments to be handed in. The oral exam consists of solving two question, one on theory and one on concrete problem, and a presentation of the  solution.

     List of Theorems whose statements and proofs are required for the exam.: (updated during the course)

1.      Proposition 2.35 (non-compactness)

2.      Theorem 3.13 (for convex sets in Hilbert spaces)

3.      Corollary 3.19. (Riesz representation)

4.      Theorem 4.1 (Banach-Steinhauss)

5.      Theorem 4.12 (Baire category)

6.      Theorem 4.28 (Closed graph)

7.      Hahn-Banach Lemma.

8.      Existence of Banach limit on l ^ \infty (small l infinity)-

9.      Banach-Alaoglu Theorem

10.  Dual space of L^1.

 

 

 

      Course requirement and plan

 

                      The learning goals of the course can be found in the course plan

      Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply           on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

    Old exams